A Compressed Sampling Receiver Based on Modulated Wideband Converter and a Parameter Estimation Algorithm for Fractional Bandlimited LFM Signals

Abstract

A modified modulated wideband converter (MWC)-based compressed sampling receiver and a parameter estimation algorithm in discrete time fractional Fourier domain (DTFrFD) to intercept and estimate the fractional bandlimited LFM signals are suggested in this work. Our proposed compressed sampling receiver has some advantages compared to the original MWC-based compressed sampling receiver. Firstly, the proposed receiver works in DTFrFD which is more effective to intercept the fractional bandlimited signals, while the original MWC-based digital receiver works in discrete time Fourier domain (DTFD). Secondly, the cross-channel signal can get better separation by the proposed digital receiver than by original MWC-based receiver because that the nonzero part of the signal’s available spectrum is narrower in DTFrFD than in DTFD. Then, with the data acquired from the digital receiver, we propose a novel parameter estimation algorithm for the intercepted fractional bandlimited LFM signal based on OMP and the differentiation spectrum in DTFrFD. The initial frequency and final frequency of the intercepted fractional bandlimited LFM signal can be extracted successfully without two-dimensional search and reconstruction, and only two iterations are needed according the algorithm. Both theoretical analysis and simulation results demonstrate that the proposed algorithm bears a relatively low complexity and its estimation precision is not only higher than both DFrFT-based search algorithm and DFrFT-based reconstruction algorithm but also close to Cramer–Rao lower bound.

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Abbreviations

MWC:

Modulated wideband converter

DFrFT:

Digital fractional Fourier transform

LFM:

Linear frequency modulation

DTFrFD:

Discrete time fractional Fourier domain

OMP:

Orthogonal matching pursuit

DTFrFT:

Discrete time fractional Fourier transform

DTFT:

Discrete time Fourier transform

DTFD:

Discrete time Fourier domain

MSE:

Mean squared error

LPI:

Low probability of intercept

SNR:

Signal-to-noise ratio

FrFT:

Fractional Fourier transform

FrFD:

Fractional Fourier domain

SFrFT:

Simplified fractional Fourier transform

FT:

Fourier transform

DSFrFT:

Digital simplified fractional Fourier transform

SFrFD:

Simplified fractional Fourier domain

AWGN:

Additive white Gaussian noise

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Acknowledgements

This research was funded by National Natural Science Foundation of China Grant No. 61271354 and Henan Province Science and Technology Key Project Grant No. 142102210431.

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Appendix A

Appendix A

Spectrum distribution characteristics of LFM signal in SFrFD A monocomponent LFM signal is defined by:

figureb

where \(f_0\) is the initial frequency. A is the amplitude of \(x\left( t\right) \) which could berandom or fixed. \(K_{\mathrm{lfm}}\) is the modulation rate. And the duration time of \(x\left( t\right) \) is \( \left[ -\frac{T_d}{2},\frac{T_d}{2}\right] \).

The SFrFT of \(x\left( t\right) \) can be calculated by

figurec

According to (S2), \({\overline{X}}_\alpha \left( u\right) \), the SFrFT of \( x\left( u\right) \), denotes the FT of another LFM signal whose modulation rate is \( K'_{\mathrm{lfm}} =K_{\mathrm{lfm}}+\cot \alpha /2 \pi \), initial frequency is \(f_0\), and the frequency interval is \(\left[ f_0-K'_{\mathrm{lfm}}\frac{T_d}{2}, f_0+K'_{\mathrm{lfm}}\frac{T_d}{2}\right] \). So the bandwidth is \(B'=K'_{\mathrm{lfm}}\cdot T_d \). It can be interpreted that the SFrFT rotates the time-frequency distribution curve of \( x\left( t\right) \) in the clockwise direction with angle \(\alpha \), so that the modulation rate and bandwidth changes.

Thus,

$$\begin{aligned} {\overline{X}}_\alpha \left( u\right)= & {} \frac{A}{\sqrt{2K'_{\mathrm{lfm}}}}\exp \left( -\frac{j\left( u-2\pi u_0\csc \alpha \right) ^2}{4\pi K'_{\mathrm{lfm}}}\right) \cdot \left\{ \left[ c\left( u_2\right) \right. \right. \\&\left. \left. + c\left( u_1\right) \right] + j\left[ s\left( u_2\right) + s\left( u_1\right) \right] \right\} \end{aligned}$$

where \( c\left( u\right) = \int _{0}^{u} \cos \left( \frac{\pi }{2}x^2 \right) \, \mathrm{d}x\) and \( s\left( u\right) = \int _{0}^{u} \sin \left( \frac{\pi }{2}x^2 \right) \, \mathrm{d}x\) is the Fresnel integral. And

figured

where \( u_0=f_0\sin \alpha \). So the amplitude spectrum of \({\overline{X}}_\alpha \left( u\right) \) is

figuree

and the phase spectrum is

figuref

Substituting \( K'_{\mathrm{lfm}}=\frac{B'}{T}\) into Eq. (S5):

figureg

According to the property associated with Fresnel integral, when \(B'T_d \gg 1\), i.e., \(K'_{\mathrm{lfm}}T_d^2 \gg 1\), \(c\left( u_2\right) \approx s\left( u_2\right) \approx \frac{1}{2}\), so

$$\begin{aligned} |{{\overline{X}}}_\alpha \left( u\right) |\approx & {} \frac{A}{\sqrt{K'_{\mathrm{lfm}}}}\cdot rect\left( \frac{u-2\pi u_0\csc \alpha }{B'}\right) \\= & {} \frac{A}{\sqrt{K_{\mathrm{lfm}}+\cot \alpha /2 \pi }}\cdot rect\left( \frac{u-2\pi u_0\csc \alpha }{B'}\right) ,\theta \left( u\right) \\\approx & {} - \frac{\pi \left( u-2\pi u_0\csc \alpha \right) ^2}{K'_{\mathrm{lfm}}}+\frac{\pi }{4}=- \frac{\pi \left( u-2\pi u_0\csc \alpha \right) ^2}{K_{\mathrm{lfm}}+\cot \alpha /2 \pi }+\frac{\pi }{4}. \end{aligned}$$

Therefore, when \(\left( K_{\mathrm{lfm}}+\cot \alpha /2 \pi \right) T_d^2\gg 1\), \(|{{\overline{X}}}_\alpha \left( u\right) |\) is approximately a rectangular spectrum.

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Li, X., Wang, H. & Luo, H. A Compressed Sampling Receiver Based on Modulated Wideband Converter and a Parameter Estimation Algorithm for Fractional Bandlimited LFM Signals. Circuits Syst Signal Process 40, 918–957 (2021). https://doi.org/10.1007/s00034-020-01507-6

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Keywords

  • Modulated wideband converter
  • Fractional Fourier transform
  • Linear frequency modulation
  • Parameter estimation
  • Orthogonal matching pursuit
  • Cramer–Rao lower bound